{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5256D2ITYEZ5YJAZD7K6Y3UQKA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7f24a1271acf87f950ef20741d42f99f7fe9b555df096474fd75d67f5709a2a4","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-05-25T21:13:58Z","title_canon_sha256":"847f3bd877126ed3645af1bebc86db0a753a32dea743e4b94c91716e289ef861"},"schema_version":"1.0","source":{"id":"1105.5153","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.5153","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1105.5153v1","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.5153","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"5256D2ITYEZ5","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5256D2ITYEZ5YJAZ","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5256D2IT","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:ff816a5032ea8826efca42a38753c6826b17669e2e2353ba63b8bbfa668bca5a","target":"graph","created_at":"2026-05-18T03:07:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We give the structure of discrete two-dimensional finite sets $A,\\,B\\subseteq \\R^2$ which are extremal for the recently obtained inequality $|A+B|\\ge (\\frac{|A|}{m}+\\frac{|B|}{n}-1)(m+n-1)$, where $m$ and $n$ are the minimum number of parallel lines covering $A$ and $B$ respectively. Via compression techniques, the above bound also holds when $m$ is the maximal number of points of $A$ contained in one of the parallel lines covering $A$ and $n$ is the maximal number of points of $B$ contained in one of the parallel lines covering $B$. When $m,\\,n\\geq 2$, we are able to characterize the case of ","authors_text":"D. Grynkiewicz, G. A. Freiman, O. Serra, Y. V. Stanchescu","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-05-25T21:13:58Z","title":"Inverse Additive Problems for Minkowski Sumsets I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5153","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e714de7f7620cf82fdff13d0ba04d7d35048a8d40873b516f7ad752474bbcda6","target":"record","created_at":"2026-05-18T03:07:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7f24a1271acf87f950ef20741d42f99f7fe9b555df096474fd75d67f5709a2a4","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-05-25T21:13:58Z","title_canon_sha256":"847f3bd877126ed3645af1bebc86db0a753a32dea743e4b94c91716e289ef861"},"schema_version":"1.0","source":{"id":"1105.5153","kind":"arxiv","version":1}},"canonical_sha256":"eebbe1e913c133dc24191fd5ec6e905008739b09e2fbd85421f7fa71bc3f2ec8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eebbe1e913c133dc24191fd5ec6e905008739b09e2fbd85421f7fa71bc3f2ec8","first_computed_at":"2026-05-18T03:07:02.394085Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:02.394085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OIoGHVe6cOy7wkDOo1DdtctZFXbMyewEVJ2NzsC+zQxwfS1X1t1S2JLtyKA3ga3FUWWGIcG9XYS1Tp3cWBCgCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:02.394597Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.5153","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e714de7f7620cf82fdff13d0ba04d7d35048a8d40873b516f7ad752474bbcda6","sha256:ff816a5032ea8826efca42a38753c6826b17669e2e2353ba63b8bbfa668bca5a"],"state_sha256":"dc3f1ddae5c3202f8445fadc3dbc7a77ed4c2ffb6a598275fd5c056188049753"}